358 Mr. A. Cayley on a Question in the 



where 



-4(«/ + Bq< -l)(*l3pq - («' + ap) ((3' + 0q) (up 4- 0qj) . 



In order that u may represent the required probability, it is 

 necessary and sufficient that it shall be 



<{; each of up, (3q, 



> „ a -hap, fi + fiq, etp + ftq; 



which implies that each of the three quantities is not less than 

 each of the two quantities ; or, what comes to the same thing, 

 /3' + /3q <£ up, u' + up <£ j3q. So that the solution is only appli- 

 cable if 



a' + «p<fc/fy, /3' + /3q<£up; 



and these conditions being satisfied, a must denote the positive 

 square root of the above-mentioned value of <r 2 . 



The values of x, y, s, t are readily obtained in terms of u ; the 

 equations in fact give 



_ up' _ /3q' s u ! (u—ftq) t _ u'(u—up) 



~7' y ~H> l~ap'{P + Pq-u)' 7~ t3q'{u' + up-uy 



where u' = l — u. And we thus have not only u, the probability 

 of the event E, but the probabilities of all the compound events 

 A'B'E', &c. I remark that the probabilities of the compound 

 events A'B', A'B, AB', AB will not be a'fi', u'/3, u/3', u/3. 



It is to be noticed that the conditions of applicability are dif- 

 ferent for the two solutions ; this, however, does not remove the 

 contradiction, as there are values of u, J3, p, q which satisfy each 

 of the two conditions. 



An interesting particular case is when 7; = 1 ; that is, when A 

 happening, E is certain to happen. 



First for my solution. The conditions are satisfied (/5, q being 

 each less than unity) if only q <£ a. The meaning of this is 

 clear ; for if B happen, then q (the probability of E) is equal to 

 u (the probability of A) plus the probability that A not happen- 

 ing, E will happen in consequence of B ; that is, q is at least 

 equal to u. 



And working out the particular case ab initio, we have 



u=\u + fxfB—Xfjb u/3, 



1=A+ (1-^/3, 



q=fjb+(l—p)\u, 



where the second equation is (1— X)(l— /-t/3)=0 ; that is, 

 1— \=0 or A=l (assuming only that ft, p, are not each of 



